Some remarks on a result of Timofeev and Khripunova

Abstract. The sum \(\sum\limits_{n\le x\atop{n\in {\cal N}_k}}\tau(n-1)\omega(n+1)\) is investigated where \(\tau(n)=\) number of divisors of \(n\), \(\omega(n)=\) number of prime divisors of \(n\), \(\Omega(n)=\) number of prime power divisors of \(n\), \(\;{\cal N}_k=\{n|\Omega(n)=k\}\).